Path integral evaluation of the one-loop effective potential in field theory of diffusion-limited reactions

The well-established effective action and effective potential framework from the quantum field theory domain is adapted and successfully applied to classical field theories of the Doi and Peliti type for diffusion controlled reactions. Through a number of benchmark examples, we show that the direct path integral calculation of the effective potential in fixed space dimension d=2 to one-loop order reduces to a small set of simple elementary functions, irrespective of the microscopic details of the specific model. Thus the technique, which allows one to obtain with little additional effort, the potentials for a wide variety of different models, represents an alternative to the standard model-dependent diagram-based calculations. The renormalized effective potential, effective equations of motion and the associated renormalization group equations are computed in d=2 spatial dimensions for a number of single species field theories of increasing complexity.